John Barsness has an article coming out soon in
Shooting Sportsman. I'm sure it will answer a lot of your questions regarding drillings.
Searching for a decent double rifle for $2,500.00 is a fool's errand. It doesn't exist.
A drilling or cape gun for around that price is certainly realistic. As has been mentioned, a drilling with an einstecklauf would give you a gun of three dimensions in one package.
This wonderful package is now in John's worthy hands. It's a drilling in 16ga 2 3/4" chambers and 6.5X57R with an added Kreighoff einstecklauf for the right barrel in 22 Mag. The scope is on Suhler claw mounts for quick removal and re-attachment, going right back to zero. The rifle will kill anything a Swede or Creed will. With the 22 Mag insert installed (a matter of seconds to do) it's a bunny & skwills gun, and without the insert it's a nimble upland bird shooter. I know of another like it that's very close to your budget. If you don't mind some heft I know of a 12/12/7X65R with 3.5-10 Leuplod in swing mount that can be had.